188 VIII DIFFERENTIAL CALCULUS

13. DIFFERENTIAL OPERATORS

Let A be an open set in R" (resp. C1), F a real (resp. complex) Banach
space; we denote by 4P)(A) the set of all/? times continuously differentiable
mappings of A into F. It is clear by (8.12.10) that 4P) (A) is a real (resp.
complex) vector space; and, more generally, (8.12.10) shows that &#\A)
(resp. 4P)(A)) is a ring, and $(/\A) a module over that ring. For any system

(a1? . . . , an) = a of integers ^0 with |a| = £ a, < p, let Ma = X^1 X? • • • XJ»
and define Da or DMa as the mapping D^1 D? • • • D*" of 4P)(A) into 4P~lal)(A).

n

A linear differential operator is a linear combination D = £ #aDa where

t = i
|a| <_/? and the <sa are continuous scalar functions defined in A; if aa = 0

for |a| > k and each aa is (;? — k) times continuously differentiable, D maps
4P)(A) linearly into 4P~fe)(A).

(8.13.1) If the operator £ <2aDa is identically 0, f/ze« eac/z of the functions aa
f,y identically 0 ?« A. a

Write D/= 0 for /(jc) = c • expC^^ + • • • + AB^), where c ^ 0 is in F
and the lt are arbitrary constants; we obtain (by Section 8.8 and (8.4.1))

+ • • • + An O = 0
identically in A, which is equivalent to £ ^(xyMJ^^ . . . , ln) = 0; for any

at

particular x e A, this implies a^x) = 0 for each a, since the Af are arbitrary.

The coefficients #a of a linear differential operator are thus uniquely
determined; the highest value of |a| such that #a ^ 0 is called the order of D.

To each polynomial P = £ £aMa of degree ^^ with constant coefficients

a

we can thus associate a linear operator DP = £*<*Da of order ^p; it is

at

clear that DPl+P2 = DPl + DP2, and it follows from (8.12.3) that if P1P2
had a total degree ^p, then DPlP2 = DPlDP2. In particular, from (8.12.7)
if follows that for fixed £y, the operator /-> D/, where

can be written
preceding condition can be replaced by the condition D2/(x0) * (*, /) < 0 for any